How to calculate Insurance Premium?
The premium is of two types: (1) Net Premium and (2) Gross Premium. The two premiums are further sub-divided into two parts: (i) single premium, and (ii) level premium. The net premium: is based on the mortality and interest rates whereas the gross premium depends upon the mortality rate, the assumed interest rate, the expenses and the bonus loading.
Single premium is paid in one lump sum while the level premium is paid periodically in installments. The level premium may be yearly, half-yearly, quarterly and monthly. Firstly, net single premium is calculated and other premiums are based on this calculation.
Net Single Premium
Net single premium is that premium which is received by the insurer in a lump sum and is exactly adequate, along with the return earned thereon, to pay the amount of claim wherever it arises whether at death or at maturity or even at surrender. It does not provide for expenses of management and for contingencies.
Steps for Calculation
1. Determine what constitutes a claim (a) death, (b) survival or (c) both.
2. Determine when claims are paid (a) at the beginning, (b) at the end, or (c) during the year.
3. Determine the number of insured.
4. Determine the duration of the policy.
5. Determine the probable number of claims per year.
6. Determine the value of claims per year.
7. Determine the number of years of interest involved and find the present value of a rupee.
8. Determine the present value of the claim for each year.
9. Determine the present value of all future claims.
10. Determine the net single premium, (i.e. present value of future claims) divided by number assumed for buying policy.
The step of premium calculation varies according to the nature of the policy which will be clear later on. When premium is calculated several questions emerged simultaneously.
Assumptions underlying Rate Computations
There are certain variables which are to be assumed at a level for calculation and alterations in premium calculation are made at later stage according to the change in the variable. The following factors are assumed while calculating the net single premium.
(i) As many policies of the given type are being issued as is the number of persons.
(ii) Premiums are collected in advance or in the beginning of the period.
(iii) All collections are immediately invested and will remain invested until money is needed for the payment of claims.
(iv) The insurer will receive an assumed rate of interest. The assumed rate should be conservative to avoid future decline in interest rate.
(v) The interest or dividend or any return of the invested funds is immediately invested for re- earning.
(vi) Mortality rate will be the same as given in the mortality table and will he uniformly distributed throughout the year.
(vii) All policies are of the same amount, say, Rs. 1,000.
(viii) Claims will be paid only at the end of the period.
These assumptions may not be totally practicable, but they are taken as for making calculation easy. The changes in assumption can be adjusted accordingly.
Calculation of Net Single Premium:
The calculation of net single premium is discussed in different types of policies.
This is the simplest type of contract whereby, payment is made only when the life assured dies within the term specified. Nothing will be paid if death does not occur during the designated term. This is also called temporary insurance. The premium is received in advance and it will not be returned if life assured survives.
The premium is paid only once in a single sum at the inception of the policy. Death claims will be paid at the end of the year in which they occur and not at the end of the term. Thus the probability of death in each year along with the present value, of the claim for each year will be calculated because the death may occur at any moment and the insurer may be required to pay. The term may be one, two, five or seven years.
Here we assume that the period of term insurance is 5 years. Before we start we assume that the (1) rate of return on investment is 3 per cent and the mortality experience will be like the one shown in the oriental 1953-54 Experience Life Table. The person is proposing at the age of 40 for the period of 5 years.
The number of details can be known from the above table we assume that each person dead will be paid Rs. 1,000. The next factor of calculation is that the insurer will earn a fixed return on the investment, therefore, only the present value of the claim should be taken as a premium. Thus, the net single premium for each year will be calculated:
Number of deaths x Amount of claims x Present value of Re. 1 = Present value of claims.
Where P stands for the present value, S for the amount of which present value is to be calculated and for the rate of interest and n for number of years for which present value is to be calculated.
Thus, the present value of claim for the first year will
be 273 x 100 x 0.971 = 265083 because the number of deaths are 273 and the total amount of claim, so, would be 2,73,000. If multiplied by factor of present value it gives present value of claim.
Thus the net single premium will be the same whether it has to be calculated on the basis of group policy or on the basis of single policy; the probability method is generally used for calculation of premium.
Net Single Premium in Whole Life Policies :
A whole life policy continues for the whole of life and promises to pay the sum assured upon the death of the insured to his beneficiary.
This policy is like the term insurance policy with only difference that instead of being limited to a definite number of years, it continues for the largest possible length of life and will certainly be paid at some time. It has been assumed in most of the mortality table that the life will continue up to 100 years.
Therefore, the calculation of premium will start from the date of commencement of risk to the 100th year. If a person has taken policy at age 45, the calculation will continue until 100th year.
The chances of death in each separate year will be multiplied by the face value of the policy and this amount is discounted by the present value for the period.
Net Single Premium in Pure Endowment Policy:
In this policy, insurer promises to pay the insured value in case the holder survives a certain fixed period. Thus the holder of 5 years pure endowment will be paid only when he survives at the end of 5 years.
The insured, cannot get possession of the money invested in a pure endowment before the expiration of the endowment period. If the insured dies during this period, the entire premium paid is forfeited.
Net Single Premium in Ordinary Endowment Policy :
Under this policy payment of claim amount is made at the survival of the term or at the death of the life assured whichever is earlier. Payment in this case is certain. Since payment is based on the death and survival, the net premium is calculated on death and survival rate.
The net single premium on the basis of death has been discussed in case of term insurance and on the basis of survival in case of pure endowment assurance. For example, we have to complete net single premium of ordinary endowment policy of 5 years, we can easily base our calculation on death and survival rates.
Net Single Premium in Double Endowment :
Under this policy, double of the amount is paid if the life assured survives at the end of the term of policy and only single amount will be paid if the death occurs within the term. Thus, it is first like ordinary endowment policy with only difference that double of the policy amount is paid if life assured survives up to the term.
Since the double of the policy amount is paid at the survival, one more premium on the basis of pure endowment is added to the premium of ordinary endowment policy. For example, double endowment policy is to be calculated of Rs. 1,000 for 5 years.'
The Net Single Premium of Ordinary Endowment + Net single premium of Pure Endowment Policies for 5 years issued at same age.
= policy for 5 years issued at the same age.
= Rs. 862.78 + 846.60 = 1709.38.
Thus, the net single premium of double endowment policy of Rs. 1,000 for 5 years will be Rs. 1.709.38. If death occurs within the term, he is paid merely Rs. 1,000 and Rs. 709.38 will be a loss to him; but if he survives up to the period, he is paid Rs. 2,006 only on payment of Rs. 1,709.38. Actuarial expression of net single premium in this case will be:
Net Single Premium for a Joint Life Policy :
Under this policy payment of claim will be made at the first death of the assured lives who may be two or more. Here, the process of calculation will be the same as has been discussed in term insurance with only difference that the probability of death is compound one. The compound probability of death is calculated by addition of the probability of deaths of one other and all of the envy aged. For example:
Compound Probability of any one of the two lives assured will be
(a) Probability of death of the younger person
(b) Probability of death of the older person
(c) Probability of death of both the persons.
It is calculated by multiplying the probability of death of each person. The compound probability may also be calculated by the following method.
Net Single Premium for Last Survival Policy :
The policy amount is payable, under this policy, only when all lives covered by the policy expire. The compound probability of all policy-holders is calculated. The calculation will continue up to the youngest life's reaching to the highest age of the mortality table, it will not stop at the first death. Thus, the compound probability will be
= Probability of death of one person x Probability of death of other person.
When the youngest son is supposed to be dead, calculation stops.